720p _ 19.1 Gate 2015
Duration: 2 min
This video lesson is available to enrolled students.
AI Summary
An AI-generated summary of this video lecture.
The video addresses a specific problem from the GATE 2015 exam regarding Finite Automata. The core question asks for the minimum number of states required in a Deterministic Finite Automaton (DFA) to accept a binary string containing an odd number of 0s, with no restriction on the number of 1s. The instructor begins by analyzing the problem statement, identifying the input alphabet as $\Sigma = \{0, 1\}$. He then proceeds to construct the DFA step-by-step on a digital whiteboard. He establishes two states, $q_0$ and $q_1$, where $q_0$ serves as the initial state representing an even count of 0s. The state $q_1$ is designed to represent an odd count of 0s. Transitions are drawn such that reading a '0' toggles the state between $q_0$ and $q_1$, effectively counting the parity of 0s. Crucially, he adds self-loops on both states for the input '1', ensuring that the presence of 1s does not affect the count of 0s. Finally, he designates $q_1$ as the accepting state because the goal is to accept strings with an odd number of 0s. Based on this construction, he concludes that exactly two states are sufficient and necessary, leading to the selection of option (B).
Chapters
0:00 – 1:39 00:00-01:39
- **00:00-01:39:** The instructor introduces the GATE 2015 question about finding minimum DFA states for strings with an odd number of 0s. He writes the alphabet $\Sigma = \{0, 1\}$ and draws a state diagram with two states, $q_0$ and $q_1$. He explains that $q_0$ is the start state (even 0s) and $q_1$ is the final state (odd 0s). He draws transitions for '0' to toggle between states and self-loops for '1' to ignore them. He circles option (B) 2 as the correct answer. The board writing includes the condition $(w)_0 = \phi \pmod 2$ initially, which seems to be a notation for counting, though the diagram clarifies the odd/even logic.
The lesson demonstrates a standard method for constructing DFAs based on parity conditions. By tracking the remainder of the count of a specific symbol modulo 2, the instructor shows that only two states are needed to distinguish between even and odd occurrences, a fundamental concept in automata theory. This approach generalizes to finding minimum states for any modulo condition $n$, requiring $n$ states.